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$$F(x) = \int{f(x)}\,dx$$

$$G(x) = \int_0^x{g(z)}\,dz$$

I am confused about the exact meaning about these functions. The second function is clear to me, $G(x)$ is just the area under the graph of $g(x)$ from $0$ to some $x$. But the first function is not so clear.

Also, why is the following considered incorrect?

$$H(x) = \int_0^x{g(x)}\,dx$$

Jason
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  • Well the first formula would only define $F$ up to an additive constant. The third is considered poor notation as the 'dummy variable' appears also as a limit. That is, the $x$ inside the integral is a formal notation indicating 'what' is being integrated. The $x$ in the limit is some fixed number. The two are unrelated, so shouldn't be denoted by the same letter. – Frank Mar 28 '15 at 00:07
  • Isn't $F(x)$ just the function obtained from doing the integral of $f(x)$?

    For example you have $F(x) = \int \frac {1}{x}dx$ and when you do the integration you'd get $\log (x) + C$ so $F(x) = \log (x) + C$.

     – mopy Mar 28 '15 at 00:09
  • $F(x)+C=\int f(x) dx \text{ where } F'=f$ – randomgirl Mar 28 '15 at 00:10

2 Answers2

2

These are good questions.

The notation $\displaystyle \int f(x) dx$ is shorthand for "an antiderivative of $f(x)$." That is, a function with the property that $F'(x) = f(x)$. Part of the depth of the fundamental theorem of calculus is that antiderivatives are also ways to calculate the area under a curve. This is remarkable - why should they be related?

The last piece you wrote, $H(x) = \displaystyle \int_0^x g(x) dx$ is terrible notation, and should not be written in front of a non-expert crowd (experts can take in notational abuse much easier than neophytes). It it far more correct to write $$ H(x) = \int_0^x g(t) dt $$ to not confuse dummy variables and the actual variable.

davidlowryduda
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0

$$ H(x) = \int_0^x{g(x)}\,dx $$ is not incorrect (just confusing for unexperienced students). This function $H$ is an antiderivative of $g$; that is $H'=g$. But usually we write $G'=g$ and $F'=f$ since it is easier to remember what is what.

Formula $$ F(x) = \int{f(x)}\,dx $$ is known as an indefinite integral and means $F'=f$.

GEdgar
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  • Arguably it is incorrect: naming bound and unbound variables with the same letter in formulae is both confusing and ambiguous: consider the case $ \int_0^x f(x,y) , dy $. – Chappers Mar 28 '15 at 00:19